Is not that simple. First you would need a flatpack daemon that manages that permission, that the application could ask for a path (e.g. by writing in a control socket) and the daemon asks the user and mounts that paths inside the container, and then inform the application that the operation is successful. But that would require modifying the application, and not in a trivial way, and it's unacceptable (you need to fork basically every application that you package for flatpack, and the filesystem access could be in various places).

A more acceptable solution would be to intercept all the filesystem related system calls, look if a path is accessible, and if not ask for the permission and either try again the system call and return the result or return to the application E_AGAIN (but is not ideal since not a lot of applications handle that correctly). But this approach would probably require a kernel module, or you can do that with eBPF but obviously you would need CAP_SYS_ADMIN capability so not really possible.

The approach of flatpack is create a container with all the paths that you know the application can access and then jump into the container. A simple solution that even doesn't require a daemon and doesn't require modifying the applications.